E(X) and Var(X) with probability calculations

5 A box contains 300 discs of different colours. There are 100 pink discs, 100 blue discs and 100 orange discs. The discs of each colour are numbered from 0 to 99 . Five discs are selected at random, one at a time, with replacement. Find

1. the probability that no orange discs are selected,
2. the probability that exactly 2 discs with numbers ending in a 6 are selected,
3. the probability that exactly 2 orange discs with numbers ending in a 6 are selected,
4. the mean and variance of the number of pink discs selected.

2. Bhim and Joe play each other at badminton and for each game, independently of all others, the probability that Bhim loses is 0.2 Find the probability that, in 9 games, Bhim loses

1. exactly 3 of the games,
2. fewer than half of the games. Bhim attends coaching sessions for 2 months. After completing the coaching, the probability that he loses each game, independently of all others, is 0.05 Bhim and Joe agree to play a further 60 games.
3. Calculate the mean and variance for the number of these 60 games that Bhim loses.
4. Using a suitable approximation calculate the probability that Bhim loses more than 4 games.

14 The random variable \(T\) follows a binomial distribution where $$T \sim \mathrm {~B} ( 16,0.3 )$$ The mean of \(T\) is denoted by \(\mu\).
14

1. \(\quad\) Find \(\mathrm { P } ( T \leq \mu )\).
   14
2. Find the variance of \(T\). \includegraphics[max width=\textwidth, alt={}, center]{f87d1b36-26db-4a0b-b9ec-d7d82a396aba-19_2488_1716_219_153}

1. A multiple-choice test consists of 25 questions, each having 5 responses, only one of which is correct.

Each correct answer gains 4 marks but each incorrect answer loses 1 mark.
Sam answers all 25 questions by choosing at random one response for each question.
Let \(X\) be the number of correct answers that Sam achieves.

1. State the distribution of \(X\) Let \(M\) be the number of marks that Sam achieves.
2. 1. State the distribution of \(M\) in terms of \(X\)
   2. Hence, show clearly that the number of marks that Sam is expected to achieve is zero. In order to pass the test at least 30 marks are required.
3. Find the probability that Sam will pass the test. Past records show that when the test is done properly, the probability that a student answers the first question correctly is 0.5 A random sample of 50 students that did the test properly was taken.
   Given that the probability that more than \(n\) but at most 30 students answered the first question correctly was 0.9328 to 4 decimal places,
4. find the value of \(n\)

In a town, 30\% of residents listen to the local radio station. Four residents are chosen at random.

1. State the distribution of the random variable X, the number of these four residents that listen to local radio. [2]
2. On graph paper, draw the probability distribution of X. [3]
3. Write down the most likely number of these four residents that listen to the local radio station. [1]
4. Find E(X) and Var (X). [3]

In a town, 30\% of residents listen to the local radio station. Four residents are chosen at random.

1. State the distribution of the random variable \(X\), the number of these four residents that listen to local radio. [2]
2. On graph paper, draw the probability distribution of \(X\). [3]
3. Write down the most likely number of these four residents that listen to the local radio station. [1]
4. Find E(\(X\)) and Var (\(X\)). [3]

20% of packets of a certain kind of cereal contain a free gift. Jane buys one packet a week for 8 weeks. The number of free gifts that Jane receives is denoted by \(X\). Assuming that Jane's 8 packets can be regarded as a random sample, find

1. P(\(X = 3\)), [3]
2. P(\(X \geqslant 3\)), [2]
3. E(\(X\)). [2]

An archer is training for the Olympics. Each of the archer's training sessions consists of 30 attempts to hit the centre of a target. The archer consistently hits the centre of the target with 79% of their attempts. It can be assumed that the number of times the centre of the target is hit in any training session can be modelled by a binomial distribution.

1. Find the mean of the number of times that the archer hits the centre of the target during a training session. [1 mark]
2. Find the probability that the archer hits the centre of the target exactly 22 times during a particular training session. [1 mark]
3. Find the probability that the archer hits the centre of the target 18 times or less during a particular training session. [1 mark]
4. Find the probability that the archer hits the centre of the target more than 26 times in a training session. [2 marks]

Abu visits his local hardware store to buy six light bulbs. He knows that 15% of all bulbs at this store are faulty.

1. State a distribution which can be used to model the number of faulty bulbs he buys. [1 mark]
2. Find the probability that all of the bulbs he buys are faulty. [1 mark]
3. Find the probability that at least two of the bulbs he buys are faulty. [2 marks]
4. Find the mean of the distribution stated in part (a). [1 mark]
5. State two necessary assumptions in context so that the distribution stated in part (a) is valid. [2 marks]

Patrick is practising his skateboarding skills. On each day, he has 30 attempts at performing a difficult trick. Every time he attempts the trick, there is a probability of 0.2 that he will fall off his skateboard. Assume that the number of times he falls off on any given day may be modelled by a binomial distribution.

1. 1. Find the mean number of times he falls off in a day. [1 mark]
   2. Find the variance of the number of times he falls off in a day. [1 mark]
2. 1. Find the probability that, on a particular day, he falls off exactly 10 times. [2 marks]
   2. Find the probability that, on a particular day, he falls off 5 or more times. [3 marks]
3. Patrick has 30 attempts to perform the trick on each of 5 consecutive days.
   1. Calculate the probability that he will fall off his skateboard at least 5 times on each of the 5 days. [2 marks]
   2. Explain why it may be unrealistic to use the same value of 0.2 for the probability of falling off for all 5 days. [1 mark]

It is known that, on average, 40% of the drivers who take their driving test at a local test centre pass their driving test. Each day 32 drivers take their driving test at this centre. The number of drivers who pass their test on a particular day can be modelled by the distribution B \((32, 0.4)\)

1. State one assumption, in context, required for this distribution to be used. [1 mark]
2. Find the probability that exactly 7 of the drivers on a particular day pass their test. [1 mark]
3. Find the probability that, at most, 16 of the drivers on a particular day pass their test. [1 mark]
4. Find the probability that more than 12 of the drivers on a particular day pass their test. [2 marks]
5. Find the mean number of drivers per day who pass their test. [1 mark]
6. Find the standard deviation of the number of drivers per day who pass their test. [2 marks]

Patrick is practising his skateboarding skills. On each day, he has 30 attempts at performing a difficult trick. Every time he attempts the trick, there is a probability of 0.2 that he will fall off his skateboard. Assume that the number of times he falls off on any given day may be modelled by a binomial distribution.

1. 1. Find the mean number of times he falls off in a day. [1 mark]
   2. Find the variance of the number of times he falls off in a day. [1 mark]
2. 1. Find the probability that, on a particular day, he falls off exactly 10 times. [2 marks]
   2. Find the probability that, on a particular day, he falls off 5 or more times. [3 marks]
3. Patrick has 30 attempts to perform the trick on each of 5 consecutive days.
   1. Calculate the probability that he will fall off his skateboard at least 5 times on each of the 5 days. [2 marks]
   2. Explain why it may be unrealistic to use the same value of 0.2 for the probability of falling off for all 5 days. [1 mark]

In a certain country 40% of the population have brown eyes. A random sample of 20 people is chosen from that population.

1. Find the expected number of people in the sample who have brown eyes. [1]
2. Find the probability that there are exactly 8 people with brown eyes in the sample. [3]
3. Find the probability that there are at least 8 people with brown eyes in the sample. [2]

In a certain country 40\% of the population have brown eyes. A random sample of 20 people is chosen from that population.

1. Find the expected number of people in the sample who have brown eyes. [1]
2. Find the probability that there are exactly 8 people with brown eyes in the sample. [3]
3. Find the probability that there are at least 8 people with brown eyes in the sample. [2]